Abstract
The popular concept of slash distribution is generalized by considering the quotient Z = X/Y of independent random variables X and Y, where X is any continuous random variable and Y has a general beta distribution. The density of Z can usually be expressed by means of generalized hypergeometric functions. We study the distribution of Z for various parent distributions of X and indicate a possible application in finance.
Highlights
A random variable Z follows a slash distribution if it can be generated as Z = X/U1/q, where X
This is the density of the standard slash distribution, which can be rewritten in various manners (usually the parameter is denoted by q in this context, see, for example, [2] (p. 233), [4] (p. 111) and [15]
The BDSL density generated can be expressed by the confluent hypergeometric function
Summary
A random variable Z follows a slash distribution if it can be generated as Z = X/U1/q , where X and U are independent random variables and U is uniformly distributed over (0,1) This term leads back to [1], where the authors considered the case in which the distribution of X, the so-called parent distribution, was normal. The slash distribution is useful when models with heavy tails are necessary to fit a real data set. This simple concept has launched a remarkable creativity among researchers. We examine the distribution of X/Y, where X may be any continuous random variable and Y is a beta distributed variable, independent from X This distribution will be called a beta divided slash distribution (BDSL distribution).
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